2 Answers
2

This is caused by the fabric, under felt and entire carpet construction to stretch, caused by people walking over it over extended periods of time.

It does not usually stretch as much as you show in your pictures. Possibly this happened because the material is sensitive to moisture and caused it to become more elastic. Moisture could be from spillage, leakage or frequent carpet washing.

Tools and techniques

If your carpet has been like that for a long time there could be irreversible damage. You could still see a crease or uneven surface after re stretching it.

Here is a nice tool (maybe you can rent one?) made specially for stretch repair.

But the more common tool is the kick stretcher and requires some elbow great!

How to do it?

You will need to completely undo one side of the carpet. Pick the easiest side to undo.

Make sure it is a side that is parallel with the crease!

In you case it looks like either the side near the tiled floor or the opposite side.

Then depends on the tool you will need to stretch the entire carper, from left to right on the length it needs to be stretched. Otherwise if you do it unevenly other creases will appear.

If possible you should put a new or check if the existing carpet gripper little hooks are still OK! These really help keep the carpet in place.

Keep on stretching and smoothing the carpet until you are satisfied, do this slowly and evenly.

Too much pressure from either tool could tear the carpet. So be careful. If in doubt, call a professional.

In case your carpet is loose

I just had another look at it looks like your carpet is more like a large rug? :)

In that case, where you do not have grippers on one side you could opt into buying some hard underlay.

Remove the entire carpet aka rug and glue the underlay to the carpet. I would not do it inside the home because it requires a special type of glue that really stinks and needs about 24 hours to bond properly, in a fairly warm and dry environment.

Thanks to the underlay, depending on what you choose your carpet will be warmer to the foot but also help keep it flat.

You need to restretch the carpet. It can be a do-it-yourself job for someone that likes to buy tools, but unless you already have them you won't save a lot. Though you might be able to rent one from a place that sells carpet retail, like (in the US) Home Depot.

Here is a page that shows how to do it, but if that link disappears, just search "carpet restretching."

What could cause this to happen? If I decide to restretch the carpet myself, do I have to rent a power stretcher, or can I simply rent a knee kicker?
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Tester101♦Jan 30 '13 at 12:58

@Tester101 - I wouldn't even bother renting - if you have a Harbor Freight nearby, they have a knee kicker for about $20 - or you could order it online.
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MT_HeadJan 30 '13 at 19:11

@MT_Head That may be true for a kicker, but I doubt you'd find a cheap power stretcher anywhere.
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Tester101♦Jan 30 '13 at 19:22

@Tester101 - Two things: 1) the definition of "power stretcher" seems to be "lever-operated rather than directly kicked", and under that definition Harbor Freight's $50 model seems to apply. 2) If you have one bump to fix, a $20 knee stretcher seems pretty reasonable; if you plan on making this a career, I'd probably go with a power stretcher - and, I must admit, I'd probably buy a nicer one than HF's. They all do the same job; the only difference is how many Advil you have to take when you're done with the job.
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MT_HeadJan 30 '13 at 21:42